Is there a way to move the to-address up (or, respectively, the date down) in the »letter« document class? With the to-address on the left and date on the right, I wish for the vertical alignment of the top row of the to-address to be the same as that of the date. Does anyone know a way of doing this?

Starting from Matthew's answer, I looked in the letter.cls file, and I found two values that I manually changed to fit my bill.

Note that this worked for my single-page, very simple letter; it is not guaranteed to work for you.

Start with this (found somewhere on the internet):

\makeatletter
\let\@texttop\relax
\makeatother

This will reduce the margin a little bit.

However, that wasn't enough for me, so I searched in letter.cls for the phrase @texttop, and found that adding the following changed the margins on the first page to what they are on the rest of the page:

I found it easier to simply alter the textwidth and margin parameters. For instance, you can insert the following commands in the preamble and change the numbers to tweak the spacing as you like. You can also insert multiple \\ at the end of the \address (or end of the \date) gives more space between the return address and the actual start of the letter.